1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
grandymaker [24]
3 years ago
15

Two streams flow into a reservoir. Let X and Y be two continuous random variables representing the flow of each stream with join

t pdf: f(x, y) = cx y(1 + y) for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3, f(x, y) = 0 otherwise. The value c, a constant, is approximately equal to:
Mathematics
1 answer:
zlopas [31]3 years ago
8 0

Answer:

c = 0.165

Step-by-step explanation:

Given:

f(x, y) = cx y(1 + y) for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3,

f(x, y) = 0 otherwise.

Required:

The value of c

To find the value of c, we make use of the property of a joint probability distribution function which states that

\int\limits^a_b \int\limits^a_b {f(x,y)} \, dy \, dx  = 1

where a and b represent -infinity to +infinity (in other words, the bound of the distribution)

By substituting cx y(1 + y) for f(x, y)  and replacing a and b with their respective values, we have

\int\limits^3_0 \int\limits^3_0 {cxy(1+y)} \, dy \, dx  = 1

Since c is a constant, we can bring it out of the integral sign; to give us

c\int\limits^3_0 \int\limits^3_0 {xy(1+y)} \, dy \, dx  = 1

Open the bracket

c\int\limits^3_0 \int\limits^3_0 {xy+xy^{2} } \, dy \, dx  = 1

Integrate with respect to y

c\int\limits^3_0 {\frac{xy^{2}}{2}  +\frac{xy^{3}}{3} } \, dx (0,3}) = 1

Substitute 0 and 3 for y

c\int\limits^3_0 {(\frac{x* 3^{2}}{2}  +\frac{x * 3^{3}}{3} ) - (\frac{x* 0^{2}}{2}  +\frac{x * 0^{3}}{3})} \, dx = 1

c\int\limits^3_0 {(\frac{x* 9}{2}  +\frac{x * 27}{3} ) - (0  +0) \, dx = 1

c\int\limits^3_0 {(\frac{9x}{2}  +\frac{27x}{3} )  \, dx = 1

Add fraction

c\int\limits^3_0 {(\frac{27x + 54x}{6})  \, dx = 1

c\int\limits^3_0 {\frac{81x}{6}  \, dx = 1

Rewrite;

c\int\limits^3_0 (81x * \frac{1}{6})  \, dx = 1

The \frac{1}{6} is a constant, so it can be removed from the integral sign to give

c * \frac{1}{6}\int\limits^3_0 (81x )  \, dx = 1

\frac{c}{6}\int\limits^3_0 (81x )  \, dx = 1

Integrate with respect to x

\frac{c}{6} *  \frac{81x^{2}}{2}   (0,3)  = 1

Substitute 0 and 3 for x

\frac{c}{6} *  \frac{81 * 3^{2} - 81 * 0^{2}}{2}    = 1

\frac{c}{6} *  \frac{81 * 9 - 0}{2}    = 1

\frac{c}{6} *  \frac{729}{2}    = 1

\frac{729c}{12}    = 1

Multiply both sides by \frac{12}{729}

c    =  \frac{12}{729}

c    =  0.0165 (Approximately)

You might be interested in
Find the median of the set of deta<br> 84, 97, 77, 31, 84, 63, 58, 72, 47, 84, 69, 94, 43, 68
blsea [12.9K]

Answer:

70.5

Step-by-step explanation:

Set of Data (Prioritized):

31, 43, 47, 58, 63, 68, <u>69, 72,</u> 77, 84, 84, 84, 94, 97

To find the median:

(69 + 72) ÷ 2

= 70.5

Therefore, the median is equal to 70.5.

6 0
2 years ago
HELP WITH THIS MATH QUESTION!!! ​I DONT KNOW HOW TO DO THIS
Fed [463]

The similarity relationship is  AC / EC = AB / ED.

The length of AB is 157.50m.

<h3>What is the length of AB?</h3>

When two triangles are similar, the ratio of the known sides of the triangles can be used to determine the length of the side of the unknown length.

The similarity relationship is : AC / EC = AB / ED

AC = 160 + 50 = 210

210 / 160 = AB / 120

AB = (210 X 120) / 160 = 157.50

To learn more about similar triangles, please check: brainly.com/question/10658464

#SPJ1

3 0
1 year ago
Tito chooses a random number between 1 and 6. What is the probability that Tito chooses a prime number?
Tanzania [10]

Answer:

he would have to pick 1,2,3 or 5...so that is 4/6 chance

if you need the percentage (im not sure if its right) it would be 76%

4 0
2 years ago
What is the sum and the product of X to the power of 2, and X
melomori [17]

Answer:

The sum is x^2 + x, the product is x^3.

Step-by-step explanation:

Adding x^2 and x leaves no like terms, so no simplification is required.

Multiplying gives x^2*x, which is x*x*x, which is then x^3

3 0
3 years ago
If two 6-sided dice are rolled, what is the<br> probability of rolling double 2s?
Komok [63]

16.7%

There are 6 ways we can roll doubles out of a possible 36 rolls (6 x 6), for a probability of 6/36, or 1/6, on any roll of two fair dice. So you have a 16.7% probability of rolling doubles with 2 fair six-sided dice.

3 0
3 years ago
Read 2 more answers
Other questions:
  • PLZ HELP ASAP!!!<br><br>suppose f(x)=x find the graph of f(x+1)
    11·1 answer
  • Is 9.1 greater then 9.099? (Im stupid)
    12·2 answers
  • What is the answer and reminder of this equation: 2,617,449 ÷ 18?
    11·1 answer
  • PLZ HELP ME I WILL GIVE BRAINLIEST TO THE CORRECT ANSWER!!!!!!!!!!!!!!!!!!!!!!
    13·1 answer
  • 1. Let f(x) be defined by the linear function graphed below and
    7·1 answer
  • #6 please help!<br> i will mark brainliest!
    8·2 answers
  • Help looks like i forgot to read a ruler
    11·1 answer
  • What is 24% as a fraction?
    12·2 answers
  • 2<br> _ y + 3 =15 What is y<br> 3
    6·1 answer
  • What is the inequality 4x-7&lt;5
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!